Theorem relcmpcmet 25371

Description: If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
relcmpcmet.1	⊢ 𝐽 = (MetOpen‘𝐷)
relcmpcmet.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
relcmpcmet.3	⊢ (𝜑 → 𝑅 ∈ ℝ+)
relcmpcmet.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)

Assertion

Ref	Expression
relcmpcmet	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Distinct variable groups:   𝑥,𝐷   𝑥,𝐽   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋

Proof of Theorem relcmpcmet

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcmpcmet.2	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
2		metxmet 24365	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋))
5		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (CauFil‘𝐷))
6		relcmpcmet.3	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑅 ∈ ℝ+)
8		cfil3i 25322	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
9	4, 5, 7, 8	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
10	3	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐷 ∈ (∞Met‘𝑋))
11		relcmpcmet.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
12	11	mopntopon 24470	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
13	10, 12	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
14		cfilfil 25320	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
15	3, 14	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
16	15	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
17		simprr 772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
18		topontop 22940	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ Top)
20		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑥 ∈ 𝑋)
21	6	rpxrd 13100	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ*)
22	21	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑅 ∈ ℝ*)
23		blssm 24449	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
24	10, 20, 22, 23	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
25		toponuni 22941	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	13, 25	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑋 = ∪ 𝐽)
27	24, 26	sseqtrd 4049	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽)
28		eqid 2740	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
29	28	clsss3 23088	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
30	19, 27, 29	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
31	30, 26	sseqtrrd 4050	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋)
32	28	sscls 23085	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
33	19, 27, 32	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
34		filss 23882	. . . . . . . 8 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑅) ∈ 𝑓 ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
35	16, 17, 31, 33, 34	syl13anc 1372	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
36		fclsrest 24053	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
37	13, 16, 35, 36	syl3anc 1371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
38		inss1 4258	. . . . . . 7 ⊢ ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fClus 𝑓)
39		eqid 2740	. . . . . . . . 9 ⊢ dom dom 𝐷 = dom dom 𝐷
40	11, 39	cfilfcls 25327	. . . . . . . 8 ⊢ (𝑓 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
41	40	ad2antlr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
42	38, 41	sseqtrid 4061	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fLim 𝑓))
43	37, 42	eqsstrd 4047	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓))
44		relcmpcmet.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
45	44	ad2ant2r 746	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
46		filfbas 23877	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
47	16, 46	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (fBas‘𝑋))
48		fbncp 23868	. . . . . . . . 9 ⊢ ((𝑓 ∈ (fBas‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
49	47, 35, 48	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
50		trfil3 23917	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
51	16, 31, 50	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
52	49, 51	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
53		resttopon 23190	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
54	13, 31, 53	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
55		toponuni 22941	. . . . . . . . 9 ⊢ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
57	56	fveq2d 6924	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
58	52, 57	eleqtrd 2846	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
59		eqid 2740	. . . . . . 7 ⊢ ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
60	59	fclscmpi 24058	. . . . . 6 ⊢ (((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp ∧ (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
61	45, 58, 60	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
62		ssn0 4427	. . . . 5 ⊢ ((((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓) ∧ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅) → (𝐽 fLim 𝑓) ≠ ∅)
63	43, 61, 62	syl2anc 583	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fLim 𝑓) ≠ ∅)
64	9, 63	rexlimddv 3167	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑓) ≠ ∅)
65	64	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
66	11	iscmet 25337	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
67	1, 65, 66	sylanbrc 582	1 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  ℝ^*cxr 11323  ℝ⁺crp 13057   ↾_t crest 17480  ∞Metcxmet 21372  Metcmet 21373  ballcbl 21374  fBascfbas 21375  MetOpencmopn 21377  Topctop 22920  TopOnctopon 22937  clsccl 23047  Compccmp 23415  Filcfil 23874   fLim cflim 23963   fClus cfcls 23965  CauFilccfil 25305  CMetccmet 25307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ico 13413  df-rest 17482  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-cmp 23416  df-fil 23875  df-flim 23968  df-fcls 23970  df-cfil 25308  df-cmet 25310

This theorem is referenced by:  cmpcmet  25372  cncmet  25375